The present invention relates to a method and apparatus for solving a diffusion equation, and in particular to a method and apparatus for solving at a high precision a diffusion equation having a diffusion source or an electric field generating source in a space to be analyzed.
In order to solve a differential equation in a data processing apparatus, a solving method is generally used to replace a first order differential df/dt with a form of finite difference
(f(t+xcex94t)xe2x88x92f(t))/xcex94txe2x80x83xe2x80x83(9)
However, the following problems have been encountered in the course of investigations toward the present invention. Namely, in order to solve, for example, a differential equation df(x, t)/dt=∂2(f(x, t)/∂2x where the diffusion coefficient D is 1, by a conventional method, it is necessary to make the finite difference (interval) xcex94t of the parameter t for keeping the calculation precision higher than a given level.
This reason resides in that divergence may otherwise occur in the course of solution of the differential equation or the calculation precision would be lowered.
If the spacial components in both sides of the equation df(x, t)/dt=∂2(f(x, t)/∂2x are expanded into Fourier-series (termed as xe2x80x9cFourier expansionxe2x80x9d), for example, f=xcexa3f1 exp(xe2x88x92k1xc2x7x), Fourier components in a form of df(t)/dt=xe2x88x92k2f(t)=xe2x88x92cf(t) is obtained. However, it is in principle necessary to make the interval xcex94t of the parameter t infinitely small since the value of c may assume from zero to infinity.
Therefore, it is hard or impossible for the conventional methods to achieve a computation at a high precision relying upon the Fourier expansion.
Accordingly, the present invent ion has been made to overcome the aforementioned problem. It is an object of the present invention to provide a totally novel method and apparatus which is capable of solving a diffusion equation at a high precision by the Fourier expansion.
According to a first aspect of the present invention, there is provided an apparatus for analyzing the diffusion state by analyzing diffusion quantity f in a space having a diffusion source therein by using a diffusion equation ∂f(r, t)/∂t=s(r, t)+D∇2f(r, t) in which a partial differential f the f with respect to time ∂f(r, t)/∂t (where r denotes spacial vector and t denotes time) is equal to the sum of the diffusion source s and a product of a diffusion coefficient D and a second order differential (∇2f) of a gradient (∇) of the f, to determine the diffusion quantity f in the space. The apparatus comprises
(a) means for Fourier-transforming spacial components with respect to f(r, t) and s(r, t) in both sides of the diffusion equation;
(b) means for modifying each of the Fourier components obtained by the Fourier-transformation into a differential equation having a form df(t)/dt=s(t)xe2x88x92cf(t) representing each spacial frequency component, where f(t) and s(t) are Fourier components of f(r, t) and s(r, t), respectively, and c includes a component of square of wave number k, and the boundary conditions of f(t) may be appropriately determined;
(c) means for solving the differential equation relating to the Fourier components by repeating the finite difference method using a given finite difference equation relating to f(t+xcex94t), f(t) and s(t) in which f(t) and s(t) are multiplied by a value of an exponential function exp(xe2x88x92cxcex94t) having a product of the c and a time interval xcex94t with a minus sign on its shoulder; and
(d) means for summing up the solutions of the differential equations relating to the Fourier components which are obtained by the finite difference equations to output the sum as a solution of the original diffusion equation.
The means for solving the differential equation relating to the Fourier components by the finite difference method is adapted to solve a difference equation represented as f(t+xcex94t)=exp(xe2x88x92cxcex94t)f(t)+(xcex94t/2)[exp(xe2x88x92cxcex94t)s(t)+s(t+xcex94t)] by the repeating method (i.e., repeating the calculus).
According to a second aspect of the present invention, there is provided a method of predicting the diffusion state. The method is characterized by the steps of:
(a) Fourier-transforming space-dependent components in a differential equation
d{right arrow over (f)}({right arrow over (r)},t)/dt={right arrow over (s)}({right arrow over (r)},t)+D∇2{right arrow over (f)}({right arrow over (r)},t)xe2x80x83xe2x80x83(3)
which is related with a position vector
{right arrow over (r)}xe2x80x83xe2x80x83(1)
and the strength of an electric field at time t
{right arrow over (f)}xe2x80x83xe2x80x83(2)
where
{right arrow over (s)}xe2x80x83xe2x80x83(4)
denotes vector in an electric field generating term, and D denotes diffusion coefficient of the electric field;
(b) and transforming each of the Fourier components into a differential equation having a form
d{right arrow over (f)}(t)/dt={right arrow over (s)}(t)xe2x88x92c{right arrow over (f)}(t)xe2x80x83xe2x80x83(5)
where f and
{right arrow over (f)},{right arrow over (s)}xe2x80x83xe2x80x83(6)
denote vector and scalar quantities, respectively, with
{right arrow over (f)}(0)=0xe2x80x83xe2x80x83(7);
(c) numerically solving the differential equations; and
(d) thereafter determining, as a solution, the sum of respective Fourier components.
In a third aspect of the present invention, the invention 5 provides a method of predicting the diffusion state in case where f and s are scalar quantities.
In accordance with the present invention, the problem that requires to make the interval (infinite difference) of the parameter t small in order to keep the calculus precision not lower than a given level as the value c increases.
The present invention is not limited to the calculation of the strength (intensity) of an electric field, but is applicable to all phenomena which may occur due to the diffusion effect such as concentration etc.
Other aspects and features of the present invention are fully mentioned in the appended claims, which disclosure is herein incorporated by reference thereto. Also other features will become apparent in the entire disclosure including the drawings.